Screwing the inscrutable, Effing the ineffable – All my postings on this site are licenced under Creative Commons BY-NC-SA

Monty Hall

Here’s an example of a counterintuitive result I came to accept.

Imagine you are a contestant on a game show whereon the host (the canonical example being Monty Hall) stands before three doors.

Behind one of the doors is a brand new car, of a brand and model which maketh you swoon. Let’s say a Tesla Roadster, just to give focus. Behind the other two doors are goats, and not very attractive goats at that; at minimum, they are goats not to be compared to a Tesla Roadster.

Monty has you pick a door out of the three. After you have picked, he gestures extravagantly toward the remaining two doors, but somewhat more extravagantly toward one than toward the other. That door flies open to reveal a goat. Monty then provides you a choice:

Do you want to stick with your original pick, or switch to the one of Monty’s doors which he hasn’t yet opened?

The question you have to ask yourself is, are your chances of winning improved by switching, by staying the same, or does it not make a gnat’s ass of difference which you choose?

I shall reveal the answer some indeterminate time after I have gotten at least one guess in the comments.

Well… You start off with a 33% chance of picking the door with the car behind it, if you decide to switch you have two doors left so you have a 50% chance of getting the car. But, that’s all to say that this game is not rigged so that no matter which door you pick you get another goat, in which case it wouldn’t make a gnat’s ass of difference.

Intuitively I was in the “makes no difference” camp, and only came to accept that it actually does after I’d run a couple of computer simulations. To be honest, I ran one and still had a hard time accepting it; I had to write one from scratch to convince myself there were no shenanigans. Not really paranoid, just stubborn!

Then, of course, I had to justify the right answer to myself! The way I finally came to accept this fully was to perform a little gedankenexperiment on the gedankenexperiment, based on the fact that Monty knows what is behind all the doors, or at least his two.

(When you picked a door, you divided the three into two groups; your single door, and Monty’s two doors, and I’ll refer to them that way.)

Obviously if the car is behind one of Monty’s doors, he’s not going to open that one. He will always open a goat door, then ask you if you want to switch. He does this in the full knowledge of what’s back there.

Now suppose instead he offered you the chance to open *both* his doors, and eliminate one *afterward* – giving you the same complete knowledge about those two doors.

Should you choose that option, you’re also going to eliminate a goat, either one of the two or the only one (assuming you don’t have a goat fetish and would actually prefer the car). Your choice in this scenario will always be the same as Monty’s in the original, so the two are functionally equivalent.

The option of picking your one door remains the same also (duh!).

Looked at that way, the choice is a lot more obvious; you’re being given a choice between looking behind two doors and taking your pick, or looking behind one door and taking Hobson’s choice!

Of course as I said I figured this out afterward, after I knew the answer already. If the situation were sprung on me, I probably would have stuck with the first door under the assumption that it made no difference. I still might have lucked into the Tesla roadster, but I would have diminished my chances severely.

Sadly, I’m unlikely to be given the opportunity anyway; even a goat would be better than nothing!